6-th order rational solutions to the KPI equation depending on 10 parameters

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6-th order rational solutions to the KPI equation depending on 10 parameters

Pierre Gaillard

To cite this version:

Pierre Gaillard. 6-th order rational solutions to the KPI equation depending on 10 parameters. Journal of Basic and Applied Research International , International Knowledge Press, 2017, 21 (2), pp.92-98.

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Families of rational solutions of order 6 to the KPI equation depending on 10

parameters.

P. Gaillard, Universit´e de Bourgogne,

Institut de math´ematiques de Bourgogne, 9 avenue Alain Savary BP 47870

21078 Dijon Cedex, France : E-mail : Pierre.Gaillard@u-bourgogne.fr

August 19, 2017

Abstract

Here we constuct rational solutions of order 6 to the Kadomtsev- Petviashvili equation (KPI) as a quotient of 2 polynomials of degree 84 in x,yandtdepending on 10 parameters. We verify that the maximum of modulus of these solutions at order 6 is equal to 2(2N+ 1)² = 338. We study the patterns of their modulus in the plane (x, y) and their evolution according time and parametersa¹,a²,a³,a⁴,a⁵,b¹,b²,b³,b⁴,b⁵. When these parameters grow, triangle and rings structures are obtained.

PACS numbers :

33Q55, 37K10, 47.10A-, 47.35.Fg, 47.54.Bd

1 Introduction

We consider the Kadomtsev-Petviashvili equation (KPI)

(4ut−6uux+uxxx)x−3uyy = 0, (1) where subscriptsx, yandt denote partial derivatives.

This equation was first proposed by Kadomtsev and Petviashvili [1] in 1970. The first rational solutions were constructed in 1977 by Manakov, Zakharov, Bordag and Matveev [4]. Other more general rational solutions of the KPI equation were found. In particular, in the works of Krichever in 1978 [5, 6], Satsuma and Ablowitz in 1979 [7], Matveev in 1979 [8], Freeman and Nimmo in 1983 [9, 10], Pelinovsky and Stepanyants in 1993 [11], Pelinovsky in 1994 [12], Ablowitz and

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Villarroel [13, 14] in 1997-1999, Biondini and Kodama [15, 16, 17] in 2003-2007.

We construct rational solutions of order N depending on 2N −2 parameters which can be written as a ratio of two polynomials of x, y and t of degree 2N(N+ 1).

The maximum of the modulus of these solutions at orderN is equal to 2(2N+ 1)². This method gives an infinite hierarchy of rational solutions of order N depending on 2N−2 real parameters. Here we construct the explicit rational solutions of order 6, depending on 10 real parameters, and the representations of their modulus in the plane of the coordinates (x, y) according to the real parametersa1,b1,a2,b2,a3,b3,a4,b4,a5,b5and timet. When the parameters grow, we obtainN(N+ 1) peaks in particular structures, such as triangles, ring, or concentric rings.

2 Rational solutions to KPI equation of order N depending on 2N −2 parameters

One defines real numbersλj such that −1< λν <1,ν = 1, . . . ,2N depending on a parameterǫwhich will be intended to tend towards 0; they can be written as

λj= 1−2ǫ²j², λN+j=−λj, 1≤j≤N, (2) The terms κν, δν, γν, τν and xr,ν are functions of λν,1 ≤ ν ≤ 2N; they are defined by the formulas :

κ_j= 2q

1−λ²_j, δj =κjλj, γj=q₁

−λj

1+λ_j,;

xr,j= (r−1) ln^γ_γ^j⁻ⁱ

j+i, r= 1,3, τj=−12iλ²_jq

1−λ²_j−4i(1−λ²_j)q 1−λ²_j, κN+j=κj, δN+j=−δj, γN+j =γ_j⁻¹,

xr,N+j =−xr,j, , τN+j =τj j= 1, . . . , N.

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eν 1≤ν ≤2N are defined in the following way : ej = 2i

P1/2M−1

k=1 ak(je)²^k⁻¹−iP1/2M−1

k=1 bK(je)²^k⁻¹ , eN+j= 2i

P1/2M−1

k=1 ak(je)²^k⁻¹+iP1/2M−1

k=1 bk(je)²^k⁻¹

, 1≤j≤N, ak, bk ∈R, 1≤k≤N−1.

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ǫν, 1≤ν≤2N are real numbers defined by :

ej= 1, eN+j= 0 1≤j≤N. (5)

The following notations are used : Xν =κνx

2 +iδνy−ix3,ν

2 −iτν

2t−ieν

2 , Yν =κνx

2 +iδνy−ix1,ν

2 −iτν

2t−ieν

2,

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for 1≤ν ≤2N, withκν,δν, xr,ν defined in (3) and parameters eν defined by (4).

We define the following functions :

ϕ4j+1,k=γ_k^4j⁻¹sinXk, ϕ4j+2,k=γ_k^4jcosXk,

ϕ4j+3,k=−γ^4j+1_k sinXk, ϕ4j+4,k=−γ_k^4j+2cosXk, (6) for 1≤k≤N, and

ϕ4j+1,N+k=γ_k^2N⁻^4j⁻²cosXN+k, ϕ4j+2,N+k=−γ^2N_k ⁻^4j⁻³sinXN+k, ϕ4j+3,N+k=−γ^2N_k ⁻^4j⁻⁴cosXN+k, ϕ4j+4,N+k=γ^2N_k ⁻^4j⁻⁵sinXN+k, (7) for 1≤k≤N.

We define the functionsψj,k for 1≤j ≤2N, 1≤k≤2N in the same way, the termXk is only replaced byYk.

ψ4j+1,k=γ^4j_k ⁻¹sinYk, ψ4j+2,k=γ_k^4jcosYk,

ψ4j+3,k=−γ_k^4j+1sinYk, ψ4j+4,k=−γ_k^4j+2cosYk, (8) for 1≤k≤N, and

ψ4j+1,N+k=γ^2N_k ^−4j−2cosYN+k, ψ4j+2,N+k=−γ^2N_k ^−4j−3sinYN+k, ψ4j+3,N+k=−γ_k^2N⁻^4j⁻⁴cosYN+k, ψ4j+4,N+k=γ^2N_k ⁻^4j⁻⁵sinYN+k, (9) for 1≤k≤N.

The rational solutions are obtained by a passage to the limit when the parameter ǫ tends towards 0. For this we make a limited expansion inǫ of all functions ϕand ψ previously defined. The arguments of the determinants thus contain only the principal parts of the limited expansions which are written as partial derivatives of these functions with respect to the parameterǫ in 0¹.

Then we get the following result as a consequence of the expression of the solutions to the KPI equation expressed in terms of Fredholm determinants and Wronskians given and proved by the author in [49]:

Theorem 2.1 The functionv defined by

v(x, y, t) =−2|det((njk)_j,k∈[1,2N])|²

det((djk)j,k∈[1,2N])² (10) is a rational solution of the KPI equation (1), where

nj1=ϕj,1(x, y, t,0),1≤j≤2N njk=^∂^2k−2_∂ǫ2k−2^ϕ^j,1(x, y, t,0),

njN+1=ϕj,N+1(x, y, t,0),1≤j≤2N njN+k= ^∂^2k_∂ǫ⁻²2k−2^ϕ^j,N+1(x, y, t,0), dj1=ψj,1(x, y, t,0),1≤j ≤2N djk =^∂^2k−2_∂ǫ2k−2^ψ^j,1(x, y, t,0),

djN+1=ψj,N+1(x, y, t,0),1≤j≤2N djN+k =^∂^2k−2_∂ǫ2k−2^ψ^j,N+1(x, y, t,0), 2≤k≤N,1≤j≤2N

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The functionsϕandψ are defined in (6),(7), (8), (9).

1This will be completely explained in a forthcoming article which is currently submitted to a review.

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3 Explicit expression of rational solutions of order 6 depending on 10 parameters

In the the following, we explicitly construct rational solutions to the KPI equation of order 6 depending on 10 parameters.

Because of the length of the expression, we cannot give it in this text.

We give patterns of the modulus of the solutions in the plane (x, y) of coordinates in functions of parametersa1,a2,a3,a4,a5, b1,b2,b3,b4, b5 and time t.

When at least, one parameter is not equal to 0, we observe the presence of 21 peak. The maximum of modulus of theses solutions is checked equal in this case N= 6 to 2(2N+ 1)²= 2×13²= 338.

Figure 1. Solution of order 6 to KPI, on the left fort= 0; in the center for t= 0,01; on the right fort= 0,1; all parameters equal to 0.

Figure 2. Solution of order 6 to KPI, on the left fort= 0,2; in the center for t= 3; on the right fort= 10; all parameters equal to 0.

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Figure 3. Solution of order 6 to KPI, on the left fora1= 10³; in the center forb1= 10³; on the right forb1= 10⁶; all parameters equal to 0 and t= 0.

Figure 4. Solution of order 6 to KPI, on the left fora2= 10⁶, sight on top; in the center forb2= 10⁶; on the right forb2= 10⁶, sight on top; all parameters

equal to 0 and t= 0.

Figure 5. Solution of order 6 to KPI, on the left fora₃= 10⁸; in the center fora3= 10⁷; on the right fora3= 10⁹; all parameters equal to 0 andt= 0.

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Figure 6. Solution of order 6 to KPI, on the left forb4= 10⁹; in the center fora5= 10⁹; on the right forb5= 10⁹; all parameters equal to 0 andt= 0.

4 Conclusion

We constructN-th order rational solutions to the KPI equation depending on 2N−2 real parameters in the caseN = 6. These solutions can be expressed in terms of a ratio of two polynomials of degree 2N(N+ 1) = 84 inx,yandt. The maximum of the modulus of these solutions is equal to 2(2N + 1)²= 338; it is important to stress that this solution which can be called lumpL6is obtained when all parameters are equal to 0 at the instantt = 0. Here we have given a complete description of rational solutions of order 6 with 10 parameters by giving explicit expressions of polynomials of these solutions.

We construct the modulus of solutions in the (x, y) plane of coordinates; differ- ent structures appear. For a givent, when one parameter grows and the other ones are equal to 0 we obtain triangular, rings or concentric rings. There are five types of patterns. Fora16= 0 orb16= 0, and other parameters equal to zero, we obtain a triangle with 21 peaks. Fora26= 0 orb26= 0, and other parameters equal to zero, we obtain three concentric rings of 5, 10, 5 peaks respectively. For a36= 0 orb36= 0, and other parameters equal to zero, we obtain three concentric rings of 7 peaks on each of them. In the case wherea46= 0 orb46= 0, and other parameters equal to zero, we obtain two rings with 9 peaks with the lumpL4in the center. In the last case wherea56= 0 or b56= 0, and other parameters equal to zero, we obtain one ring with 11 peaks with the lumpL4 in the center.

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Appendix : Because of the length of the complete expression, we only give in this appendix the explicit expression of the rational solution of order 6 to KPI equation without parameters. They can be written as

v(x, y, t) =−2|n(x, y, t)|² (d(x, y, t))² withX= 2x,Y = 2y,T = 4t,

d(2x,4y,4t) =X42 + 146368466364027986997712983000T28−163785212871891660328043019543750T18−

1743258987219416166292162033125000T14 +25162642040415238402427879381250T24−17796475908040409761437226372500T22− 2259558449088511852089565548750T26−14382518238064346771583T40−16261155226563941719711350T36 +109418989131512359209T42 + 687164760263074345753410T38−1464804007245369414761534775T32 + 12196285846533687111121049400T30−

709906882928454149923781466862500T20 +628004925677254432643853460453125T10+98684940687917333013467983828125T8− 5261801662602586032097211718750T6−171395180519593084257508593750T4 + 9886311444358219573828125T2−

1196227258580599721074391193843750T16 +2832136699793899349608144455375000T12 +28006548001014786328125+

206270951783460976572676425T34 + (77071629600T+ 2101471192800T3 + 39070080iY9T+ 15471751680iY7T3 + 1462080533760iY5T5 +46368839784960iY3T7 +452096187903360iY T9−2969326080iY7T−735315356160iY5T3− 37911491735040iY3T5−480771654612480iY T7 +23638728960iY5T+2402127705600iY3T3 +44760184761600iY T5− 2895782400iY3T−2549863411200iY T3−110611267200iY T−2055567326400T3Y4−69519364326720T5Y2 + 450848160T Y8 +146778065280T3Y6 +10972016637120T5Y4 +259299433008000T7Y2−1953504T Y10−966984480T3Y8− 121840044480T5Y6−5796104973120T7Y4−113024046975840T9Y2−34682709600T Y2 + 14663073600T Y4 + 842761584000T3Y2−11713947840T Y6−758288606074272T11 + 1695360704637600T9−549756022052160T7 + 37592494632000T5 )X31 + 189T2Y40 + 17010T4Y38 + 969570T6Y36 + 39267585T8Y34 + 1201588101T10Y32 +

28838114424T12Y30 +556163635320T14Y28 +8759577256290T16Y26+113874504331770T18Y24+1229844646783116T20Y22 + 11068601821048044T22Y20 +83014513657860330T24Y18 +517244277406668210T26Y16 +2660113426662865080T28Y14 + 11172476391984033336T30Y12 +37707107822946112509T32Y10 +99812932472504415465T34Y8 +199625864945008830930T36Y6 + 283678860711328338690T38Y4 +255310974640195504821T40Y2 +(21Y2 +7749T2−84iY−63)X40+(−2520T Y2−

309960T3 + 10080iY T + 10920T)X39 + (210Y4 + 147420T2Y2 + 9066330T4−1680iY3−589680iY T2− 6300Y2−835380T2 +5040iY−3150)X38 +(−18158175 + 774396043785T8−646379353620T6 +36433633950T4− 1003760100T2−90493200iY5T2−10044745200iY3T4−235047037680iY T6 +1927346400iY3T2 +72427899600iY T4 + 315932400iY T2 +2273840100T2Y2−626705100T2Y4−48241210500T4Y2 +7541100T2Y6 +1255593150T4Y4 + 58761759420T6Y2 + 6747300Y2 + 9591750Y4−873180Y6 + 5985Y8−95760iY7 + 4127760iY5−4762800iY3 + 17463600iY)X34 + (1601293050T+ 38091551400T3 + 9767520iY7T+ 3076768800iY5T3 + 204912802080iY3T5 + 3424971120480iY T7−435425760iY5T−70279876800iY3T3−1693226838240iY T5 +782157600iY3T+17327066400iY T3− 1565373600iY T+21901735800T3Y4 +1038045115800T5Y2−610470T Y8−256397400T3Y6−25614100260T5Y4− 856242780120T7Y2−462785400T Y2−1078906500T Y4−98578053000T3Y2 +90263880T Y6−8776488496230T9 +

10560327621480T7−1225262448900T5 )X33 +(−973099575+86887236112677T10−144812060187795T8 +29491102213650T6− 1293181762950T4−71229162375T2−483492240iY7T2−76150027800iY5T4−3381061234320iY3T6−42384017615940iY T8 + 22266090000iY5T2 +1856991906000iY3T4 +31497254656560iY T6−55723172400iY3T2−1294550472600iY T4 +

73809111600iY T2 +30218265T2Y8 +6345835650T4Y6 +422632654290T6Y4 +10596004403985T8Y2 +2985084900T2Y2 + 58675116150T2Y4 + 3008928072150T4Y2−4527438300T2Y6−556763580450T4Y4−18017497367100T6Y2 + 410493825Y2−79975350Y4 +117892530Y6−4674915Y8 +20349Y10−406980iY9 +30587760iY7−222105240iY5− 26989200iY3 + 1234755900iY)X32 + (−335047977180000T−4517956882908000T3 + 291903248448000iY7T3 + 18234200825406720iY5T5 + 409672089133171200iY3T7 + 3180008423953468800iY T9−5690110809600iY7T−

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